Found problems: 85335
Let \( \triangle ABC \) be an acute triangle with orthocenter \( H \), and let \( M \) be the midpoint of \( BC \). Let \( P \) be the foot of the perpendicular from \( H \) to \( AM \). Prove that \( AM \cdot MP = BM^2 \).
In a football tournament with $n\geq 2$ teams each two played a match. For a won match the victor gets 2 points and for a draw each one gets 1 point. In the final results there weren’t two teams with equal amount of points.
It turned out that because of a mistake each match that was written in the results as won was actually a draw and each one that was written as draw was actually won. In the new ranking there were also no two teams with the same amount of points.
Find all n for which it is possible for the two rankings to be opposite of each other, that is the first team in the first ranking is actually the last one, the second team is pre-last and so on.
There are eight seats at a round table. Six adults $A_1, \ldots, A_6$ and two children sit around the table. The two children are not allowed to six next to each other. All the seating configurations where the children are not seated next to each other are equally likely. What is the probability that the adults $A_1$ and $A_2$ end up sitting next to each other?\[
\mathrm a. ~4/15\qquad \mathrm b. ~2/7 \qquad \mathrm c. ~2/9 \qquad\mathrm d. ~1/3\qquad\mathrm e. ~1/5\qquad\]
How many different positive integers divide $10!$ ?
For a complex number $ z\equal{}x\plus{}iy$ we denote by $ P(z)$ the corresponding point $ (x,y)$ in the plane. Suppose $ z_1,z_2,z_3,z_4,z_5,\alpha$ are nonzero complex numbers such that:
$ (i)$ $ P(z_1),...,P(z_5)$ are vertices of a complex pentagon $ Q$ containing the origin $ O$ in its interior, and
$ (ii)$ $ P(\alpha z_1),...,P(\alpha z_5)$ are all inside $ Q$.
If $ \alpha\equal{}p\plus{}iq$ $ (p,q \in \mathbb{R})$, prove that $ p^2\plus{}q^2 \le 1$ and $ p\plus{}q \tan \frac{\pi}{5} \le 1$.
There are $n$ non-coplanar points in space. Prove that there exists a circle exactly passes through three points of them.
Straight rod of 2 meter length is cut into $N$ sticks. The length of each piece is an integer number of centimeters. For which smallest $N$ can one guarantee that it is possible to form the contour of some rectangle, while using all sticks and not breaking them further?
(Author: A. Magazinov)
find all functions from the nonegative integers into themselves, such that: $2f(m^2+n^2)=f^2(m)+f^2(n)$ and for $m\geq n$ $f(m^2)\geq f(n^2)$.
$9.$ What is the greatest possible perimeter of a right-angled triangle with integer side lengths if one of the sides has length $12 ?$
Determine all pairs $(a,b) \in \mathbb{C} \times \mathbb{C}$ of complex numbers satisfying $|a|=|b|=1$ and $a+b+a\overline{b} \in \mathbb{R}$.
Prove that if $p$ is an odd prime, then $p^2(p^2 -1999)$ is divisible by $6$ but not by $12$.
Let $m$ and $n$ be two positive integers, $m, n \ge 2$. Solve in the set of the positive integers the equation $x^n + y^n = 3^m$.
Let $a, b, c$ be real numbers with the property as $ab + bc + ca = 1$. Show that:
$$\frac {(a + b) ^ 2 + 1} {c ^ 2 + 2} + \frac {(b + c) ^ 2 + 1} {a ^ 2 + 2} + \frac {(c + a) ^ 2 + 1} {b ^ 2 + 2} \ge 3 $$.
In the triangle $ ABC $ on the sides $ BC $, $ CA $, $ AB $, the points $ D $, $ E $, $ F $ are chosen respectively in such a way that $$
BD \colon DC = CE \colon EA = AF \colon FB = k,$$
where $k$ is a given positive number. Given the area $ S $ of the triangle $ ABC $, calculate the area of the triangle $ DEF $
Let $ f(x)$and $ g(y)$ be two monic polynomials of degree=$ n$ having complex coefficients.
We know that there exist complex numbers $ a_i,b_i,c_i \forall 1\le i \le n$, such that
$ f(x)\minus{}g(y)\equal{}\prod_{i\equal{}1}^n{(a_ix\plus{}b_iy\plus{}c_i)}$.
Prove that there exists $ a,b,c\in\mathbb{C}$ such that
$ f(x)\equal{}(x\plus{}a)^n\plus{}c\text{ and }g(y)\equal{}(y\plus{}b)^n\plus{}c$.
A set of points on the plane is [i]antiparallelogram [/i] if any four points of the set are [b]not[/b] vertices of a parallelogram. Prove that for any set of $2023$ points on the plane, [b]no[/b] three of them are collinears, there exists a subset of $17$ points, such that this subset is antiparallelogram.
a) Given a chain of $60$ links each weighing $1$ g. Find the smallest number of links that need to be broken if we want to be able to get from the obtained parts all weights $1$ g, $2$ g, . . . , $59$ g, $60$ g? A broken link also weighs $1$ g.
b) Given a chain of $150$ links each weighing $1$ g. Find the smallest number of links that need to be broken if we want to be able to get from the obtained parts all weights $1$ g, $2$ g, . . . , $149$ g, $150$ g? A broken link also weighs $1$ g.
The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In five years Jack will be twice as old as Bill will be then. What is the difference in their current ages?
$ \textbf{(A)}\ 9 \qquad
\textbf{(B)}\ 18 \qquad
\textbf{(C)}\ 27 \qquad
\textbf{(D)}\ 36 \qquad
\textbf{(E)}\ 45$
Tow circumferences of radius $R_1$ and $R_2$ are tangent externally between each other. Besides that, they are both tangent to a semicircle with radius of 1, as shown in the figure. (Diagram is in the attachment)
a) If $A_1$ and $A_2$ are the tangency points of the two circumferences with the diameter of the semicircle, find the length of $\overline{A_1 A_2}$.
b) Prove that $R_{1}+R_{2}=2\sqrt{R_{1}R_{2}}(\sqrt{2}-\sqrt{R_{1}R_{2}})$.
Find all real $\alpha$ s.t.
\[ [ \sqrt{n + \alpha} + \sqrt{n} ] = [ \sqrt{4n+1} ] \] holds for all natural numbers $n$
The number $a_n$ is formed by writing in succession, without spaces, the numbers $1, 2, ..., n$ (for example, $a_{11} = 1234567891011$). Find the smallest number t such that $11 | a_t$.
Let $k$ be a fixed positive integer. The $n$th derivative of $\tfrac{1}{x^k-1}$ has the form $\tfrac{P_n(x)}{(x^k-1)^{n+1}}$, where $P_n(x)$ is a polynomial. Find $P_n(1)$.
Do there exist a rectangle that can be partitioned into a regular hexagon with side length $1$, and several right triangles with side lengths $1, \sqrt3 , 2$?
Let $n>1$ be an integer and $a,b,c$ be three complex numbers such that $a+b+c=0$ and $a^n+b^n+c^n=0$. Prove that two of $a,b,c$ have the same magnitude.
[i]Evan O'Dorney.[/i]
Any positive integer $n$ can be written in the form $n = 2^b(2c+1)$. We call $2c+1$ the[i] odd part[/i] of $n$. Given an odd integer $n > 0$, define the sequence $ a_0, a_1, a_2, ...$ as follows: $a_0 = 2^n-1, a_{k+1} $ is the [i]odd part[/i] of $3a_k+1$. Find $a_n$.